INTEGRABLE SYSTEMS.

INTEGRABLE SYSTEMS.

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This book illustrates the powerful interplay between topological, algebraic and complex analytical methods, within the field of integrable systems, by addressing several theoretical and practical aspects. Contemporary integrability results, discovered in the last few decades, are used within differe...

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Bibliographic Details
Main Author: LESFARI, AHMED
Format: Electronic eBook
Language:English
Published: [S.l.] : JOHN WILEY, 2022.
Subjects:
Integral equations.
Hamiltonian systems.
Online Access:CONNECT
Table of Contents:
  • Cover
  • Half-Title Page
  • Dedication
  • Title Page
  • Copyright Page
  • Contents
  • Preface
  • Chapter 1. Symplectic Manifolds
  • 1.1. Introduction
  • 1.2. Symplectic vector spaces
  • 1.3. Symplectic manifolds
  • 1.4. Vectors fields and flows
  • 1.5. The Darboux theorem
  • 1.6. Poisson brackets and Hamiltonian systems
  • 1.7. Examples
  • 1.8. Coadjoint orbits and their symplectic structures
  • 1.9. Application to the group SO(n)
  • 1.9.1. Application to the group SO(3)
  • 1.9.2. Application to the group SO(4)
  • 1.10. Exercises
  • Chapter 2. Hamilton-Jacobi Theory
  • 2.1. Euler-Lagrange equation
  • 2.2. Legendre transformation
  • 2.3. Hamilton's canonical equations
  • 2.4. Canonical transformations
  • 2.5. Hamilton-Jacobi equation
  • 2.6. Applications
  • 2.6.1. Harmonic oscillator
  • 2.6.2. The Kepler problem
  • 2.6.3. Simple pendulum
  • 2.7. Exercises
  • Chapter 3. Integrable Systems
  • 3.1. Hamiltonian systems and Arnold-Liouville theorem
  • 3.2. Rotation of a rigid body about a fixed point
  • 3.2.1. The Euler problem of a rigid body
  • 3.2.2. The Lagrange top
  • 3.2.3. The Kowalewski spinning top
  • 3.2.4. Special cases
  • 3.3. Motion of a solid through ideal fluid
  • 3.3.1. Clebsch's case
  • 3.3.2. Lyapunov-Steklov's case
  • 3.4. Yang-Mills field with gauge group SU(2)
  • 3.5. Appendix (geodesic flow and Euler-Arnold equations)
  • 3.6. Exercises
  • Chapter 4. Spectral Methods for Solving Integrable Systems
  • 4.1. Lax equations and spectral curves
  • 4.2. Integrable systems and Kac-Moody Lie algebras
  • 4.3. Geodesic flow on SO(n)
  • 4.4. The Euler problem of a rigid body
  • 4.5. The Manakov geodesic flow on the group SO(4)
  • 4.6. Jacobi geodesic flow on an ellipsoid and Neumann problem
  • 4.7. The Lagrange top
  • 4.8. Quartic potential, Garnier system
  • 4.9. The coupled nonlinear Schrödinger equations
  • 4.10. The Yang-Mills equations.
  • 4.11. The Kowalewski top
  • 4.12. The Goryachev-Chaplygin top
  • 4.13. Periodic infinite band matrix
  • 4.14. Exercises
  • Chapter 5. The Spectrum of Jacobi Matrices and Algebraic Curves
  • 5.1. Jacobi matrices and algebraic curves
  • 5.2. Difference operators
  • 5.3. Continued fraction, orthogonal polynomials and Abelian integrals
  • 5.4. Exercises
  • Chapter 6. Griffiths Linearization Flows on Jacobians
  • 6.1. Spectral curves
  • 6.2. Cohomological deformation theory
  • 6.3. Mittag-Leffler problem
  • 6.4. Linearizing flows
  • 6.5. The Toda lattice
  • 6.6. The Lagrange top
  • 6.7. Nahm's equations
  • 6.8. The n-dimensional rigid body
  • 6.9. Exercises
  • Chapter 7. Algebraically Integrable Systems
  • 7.1. Meromorphic solutions
  • 7.2. Algebraic complete integrability
  • 7.3. The Liouville-Arnold-Adler-van Moerbeke theorem
  • 7.4. The Euler problem of a rigid body
  • 7.5. The Kowalewski top
  • 7.6. The Hénon-Heiles system
  • 7.7. The Manakov geodesic flow on the group SO(4)
  • 7.8. Geodesic flow on SO(4) with a quartic invariant
  • 7.9. The geodesic flow on SO(n) for a left invariant metric
  • 7.10. The periodic five-particle Kac-van Moerbeke lattice
  • 7.11. Generalized periodic Toda systems
  • 7.12. The Gross-Neveu system
  • 7.13. The Kolossof potential
  • 7.14. Exercises
  • Chapter 8. Generalized Algebraic Completely Integrable Systems
  • 8.1. Generalities
  • 8.2. The RDG potential and a five-dimensional system
  • 8.3. The Hénon-Heiles problem and a five-dimensional system
  • 8.4. The Goryachev-Chaplygin top and a seven-dimensional system
  • 8.5. The Lagrange top
  • 8.6. Exercises
  • Chapter 9. The Korteweg-de Vries Equation
  • 9.1. Historical aspects and introduction
  • 9.2. Stationary Schrödinger and integral Gelfand-Levitan equations
  • 9.3. The inverse scattering method
  • 9.4. Exercises.
  • Chapter 10. KP-KdV Hierarchy and Pseudo-differential Operators
  • 10.1. Pseudo-differential operators and symplectic structures
  • 10.2. KdV equation, Heisenberg and Virasoro algebras
  • 10.3. KP hierarchy and vertex operators
  • 10.4. Exercises
  • References
  • Index
  • Other titles from iSTE in Mathematics and Statistics
  • EULA.

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